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A paper by Gabor Etesi was published that purports to solve a major outstanding problem:

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking Journ. Math. Phys. 56, 043508-1-043508-21 (2015) journal version, current arXiv version.

Since this is obviously an important and groundbreaking result (if true), published in a physical journal, I am interested whether it is accepted by mathematicians.

David Roberts
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Misha Verbitsky
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    This is the wrong forum for your question. This kind of question falls in my view to discussing of preprints. If you find a particular error and want confirmation of that error, you might ask about that specific detail. I would avoid editorializing or commenting on the quality of the paper; it should be enough to get people to recognize a serious defect in the paper, if one exists. Gerhard "Treat All Papers With Respect" Paseman, 2015.06.24 – Gerhard Paseman Jun 24 '15 at 22:28
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    I do not think it is the wrong forum. This is not a preprint, it is a peer-reviewed and published paper claiming to solve a very famous and long-standing problem. It is completely natural to wonder whether it is actually correct or not. At any rate, I agree that the question could be improved. – Francesco Polizzi Jun 24 '15 at 22:38
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    At this current writing, the question is not of the quality I would like to see on MathOverflow. It renders a subjective opinion which may be correct but I will not know without doing some extensive effort. Francesco, If I recall a preprint of yours from ten years ago, and notice a recent publication of a similar paper by you, how appropriate is it for me to publicly claim "Ten years ago it needed a lot of work; can it possibly have gotten publishable?" MathOverflow is not for critiquing papers; it is for answering questions. Gerhard "Let's Not Talk About Publishers" Paseman, 2015.06.24 – Gerhard Paseman Jun 24 '15 at 22:44
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    Why don't you read the newest version yourself and see if the errors are still there? The paper has been updated a lot of times in those 10 years. – Tobias Kildetoft Jun 24 '15 at 22:46
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    Using arXiv trackbacks, one finds other questions and comments on MathOverflow that may be useful in gauging the quality of this paper. If I were the author, I would prefer someone pointing out a specific problem in the current version rather than pointing to older versions and suggesting that insufficient improvements had been made. Gerhard "Treat Authors With Respect Too" Paseman, 2015.06.24 – Gerhard Paseman Jun 24 '15 at 23:01
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    I'm voting to close this question as off-topic because: I agree with Gerhard that it is inappropriate to discuss the flaws in a paper in a public forum. – Joseph O'Rourke Jun 24 '15 at 23:39
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    Previous discussion at http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere – Gerry Myerson Jun 25 '15 at 01:04
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    I honestly don't understand these comments. The person who posted the question is one of the leading experts in hyperkahler geometry worldwide. He is trying to understand whether the community has changed opinion regarding a paper he himself has already read and found incorrect a long time ago. It would be beneficial for the entire mathematical community to understand the status of the claimed result. (that being said, one might encourage the OP to rephrase the question so that it appears more polite and kind towards the author) – pro Jun 25 '15 at 01:04
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    The basic intent, of finding out whether the paper is correct, is indeed of interest. That does not mean that it is appropriate for MathOverflow, just as conjectural discussion of Mochizuki's proof of the ABC conjecture and other examples are not appropriate. The previous questions address some of the mathematics and reasons for why the statement might be true or false. The question here does not, and can be viewed as damaging to the author and (more importantly) to MathOverflow. I don't challenge the intent so much as the form. Gerhard "Maybe I Should Just Edit" Paseman, 2015.06.24 – Gerhard Paseman Jun 25 '15 at 01:14
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    @GerhardPaseman I disagree and I think you are missing the point. I would also be against some random dude trying to undermine a random paper written by some random dude (or worse, a random grad student). Here we are talking about a paper claiming to prove a big conjecture. If correct, this deserves to be published in a top mathematical journal, not a physics journal. We are talking about a paper which has had a bad reputation for ten years now. (continued below) – pro Jun 25 '15 at 01:50
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    If the experts in the field (such as MV) don't buy the paper, how does it even stand a chance? I can only see two outcomes here. Either someone answers confirming there are still flaws in the proof, and we are left with the status quo, where the reputation of the author is unchanged (same outcome if no answers are received). Or someone answers vouching for the paper. Now wouldn't this be great? Someone like MV will then actually take time to read the paper, if he finds it correct he'll also vouch for it. And, there you go, the author has the reputation he very well deserves! – pro Jun 25 '15 at 01:51
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    The question raised is a valid one, though it could be slightly edited to sound less confrontational. Closing this discussion, would do a great disservice to the math community and smacks of censorship. – Liviu Nicolaescu Jun 25 '15 at 07:40
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    Better to also give a link to the published version, http://dx.doi.org/10.1063/1.4918540, if an when this question is edited for improvement. – David Roberts Jun 25 '15 at 11:33
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    @LiviuNicolaescu: I agree that this question should be open, but comparing its closure to "censorship" is ridiculous and is only likely to derail the discussion. The fact that MO is not considered to be the correct forum for certain questions is not censorship. – Eric Wofsey Jun 25 '15 at 12:14
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    I've opened up a discussion at meta: http://meta.mathoverflow.net/questions/2328/on-discussion-of-published-papers-at-mo Please consider upvoting this comment for visibility. – Todd Trimble Jun 25 '15 at 12:49
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    I have informed Gabor Etesi about this MO discussion. I have also encouraged him to comment on the differences between the first version of his paper from 10 years ago, and the current published version. – André Henriques Jun 25 '15 at 17:18
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    I apologize for being overly blunt (and thanks to those who pointed this out). The question is edited. For the record, I said "I have read it when it first appeared in arxiv 10 years ago, there were not very subtle errors then. Is it still wrong? " – Misha Verbitsky Jun 25 '15 at 20:38
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    Misha, I don't really want to push this back towards you, but there probably aren't a lot of other people who are in a better position than you to judge the correctness of this paper. Especially since you already looked at an earlier version. – Deane Yang Jun 25 '15 at 22:49
  • @AndréHenriques that I feel is a productive approach, as long as it it contained to addressing specifics, and not allowed to run on and on -- this isn't a forum for convincing people the published version is correct. – David Roberts Jun 26 '15 at 01:44
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    Professor Verbitsky: I don't know whether you've been following the meta discussion, but there seems to be growing consensus that the question might still be improved if you could perhaps relate, within the body of this question, some of the specific mathematical concerns you had which originally led to asking this question (concerns you alluded to before the edits). Many believe it is best practice for MathOverflow, and perhaps all the more so when asking questions about correctness of a paper, to pinpoint precise mathematical issues. Many thanks for your understanding. – Todd Trimble Jun 26 '15 at 02:07
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    The question as currently formulated claims that JMP is a "physics" not a "mathematics" journal (and not both). Given that all five editor's picks currently on the journal home page use a definition-lemma-theorem format, the editors and authors (of which I am not one) may disagree with this characterisation. Since the referees of even the most highly reputable non-physical mathematics journals are not infallible, I suspect the issue of the journal is a red herring and that the question would be better focused on specifics of the mathematical substance. –  Jun 26 '15 at 07:46
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    I agree with the last comment of Todd Trimble, and I would like to see Misha Verbitsky comment/expand on the specific errors that were present in the earlier versions. I would also like him to comment on whether or not those errors have been addressed/corrected in the present published version. I shall now cast the last closing vote. I hope that the question will be edited, at which point, I will vote to reopen. – André Henriques Jun 26 '15 at 10:02
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    I have communicated with Etesi. He claims that the published paper is essentially all new material, and that it has very little in common with the previous (flawed) preprint. – André Henriques Jun 26 '15 at 10:42
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    @AndréHenriques: Note, though, that the web page for the current arXiv version states that this version (Version 7) is the published version. – Robert Bryant Jun 27 '15 at 15:22
  • @RobertBryant but the question originally said OP read it 10 years ago. So it ought to have been another version, the first likely. I assume the reply is about that version not the current one. –  Jun 27 '15 at 18:21
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    I am quite worried by the fact that the subbundle $H\subset TG_2$ that the author defines is not integrable. Just before (13) the author says that the Levi-Civita connection $\nabla_0$ preserves the splitting $TG_2=V\oplus H$, but this would certainly imply that $H$ is closed under Lie bracket. – YangMills Jun 29 '15 at 08:27
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    @YangMills based on the discussion in the meta thread, a question about this specific claim in the paper would be on-topic for MO, and contacting the author and asking them about it might be a good idea. – Will Sawin Jul 07 '15 at 13:43
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    Etesi has posted a new preprint that extracts (most of) the purely mathematical content from his published paper: http://arxiv.org/abs/1509.02300 (In it he explicitly thanks the MO community for discussions.) – David Roberts Sep 09 '15 at 03:23
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    FYI: A proof was posted by Atiyah on arvix Five days ago! – Arctic Char Oct 31 '16 at 06:35
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    According to some experts I've consulted, there is a lot of skepticism of Atiyah's proof. – Deane Yang Oct 31 '16 at 18:02
  • only time will tell – janmarqz Oct 31 '16 at 22:37

2 Answers2

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I am Gabor Etesi, the author of the current paper "Complex structure on the six dimensional sphere from a spontaneous symmetry breaking", Journ. Math. Phys. 56, 043508-1-043508-21 (2015). First of all I would like to thank for the interest in my work on this classical problem. Because I have been asked by Andre Henriques, hereby I confirm that this published 2015 version is indeed completely independent of the wrong and withdrawn 2005 version available on arXiv. Therefore it is absolutely unnecessary to spend time with that version.

This new published 2015 version is self-contained and constructs a complex structure on $S^6$ using the following two simple observations:

(i) A complex structure on a complex manifold can always be re-interpreted as a spontaneously broken classical vacuum solution in a non-linear version of a Yang-Mills-Higgs theory formulated on the underlying manifold. This is because an almost complex tensor field $J$ is mathematically the same as a Higgs field $\Phi$. Although the terminology comes from theoretical physics, the corresponding mathematical structures are well-defined;

(ii) A complex structure on $S^6$ is then constructed as the Fourier expansion of the usual Samelson complex structure (regarded as spontaneously broken vacuum solution in the sense of item (i) above) on the exceptional Lie group $G_2$. The mathematical theory of this Fourier expansion is itself very useful and is contained in the text.

Please note that this is a freshly new approach to this old problem, apparently without a predecessor.

Finally, I would like to kindly ask everybody to read the paper first, before sending negative comments. I am ready to explain some details of the construction however please understand that I cannot take part in an infinitely long discussion. (Recently I have been working on different stuffs.)

Thanks again,

Gabor

Myshkin
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vonTuzzentahler
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Here is an answer to the question of YangMills (thank you!) regarding the subbundle $H \subset TG_2$:

I did not claim in the text that the subbundle $H \subset TG_2$ should be integrable in any sense. What I only need is the formal fact that the functional in eq. (15) vanishes on the constructed triple $(\nabla_H,J_H,g_H)$ as can be verified by a direct calculation. However I acknowledge that YangMills was right and the Levi--Civita connection does not preserve $H\subset TG_2$ as it was written in the text. For a corrected version please visit:

http://www.math.bme.hu/~etesi/s6-spontan.pdf

Neverthekess after this observation the proof proceeds as follows: $J_H$ is Fourier expanded and the corresponding ground mode, denoted by $J$ in the text after eq. (21), descends to $S^6$. The very important subtlety however is this (explained carefully in Section III): in our situation (i.e., Fourier expanding general sections of general vector bundles), there is NO canonical way to perform a Fourier expansion. Instead there is "moduli space" of possible Fourier expansions resulting in inequivalent ground modes. This is because doing fiberwise integration does NOT commute with gauge transformations hence Fourier expanding a gauge transformed (on $H$) section is NOT the same as gauge transforming (on $TS^6$) the ground mode of a Fourier expanded section. I construct in Lemma 5.1 a distinguished "$\alpha$-twisted" Fourier expansion of whose ground mode $J$ coincides with $J_H$ itself.

I think that most of the concerns and uncertainty about the published version is related with the historical remark that the "relationship" between Yang--Mills theory (mathematically invented in the 1980's) and classical complex manifold theory, more precisely the Kodaira--Spencer deformation theory (invented in the 1950-60's) is not fully clarified. By this I mean that apparently the action of the gauge group on an (almost) complex manifold $(M,J)$ is dubious: it can describe both just a symmetry transformation of $(M,J)$ or an effective deformation of $(M,J)$. But these certainly should be carefully distinguished. My suggestion is formulated in the "Principle" of Section II (but this point might require a more conceptional and less ad hoc work, I agree).

vonTuzzentahler
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